16 mathematics of normal distributions

Excursions in Modern Mathematics, 7e: 16.2 - 2Copyright © 2010 Pearson Education, Inc.

16 Mathematics of Normal Distributions

16.1 Approximately Normal Distributions of Data

16.2 Normal Curves and Normal Distributions

16.3 Standardizing Normal Data

16.4 The 68-95-99.7 Rule

16.5 Normal Curves as Models of Real-Life Data Sets

16.6 Distribution of Random Events

16.7 Statistical Inference


The study of normal curves can be traced back to the work of the great German mathematician Carl Friedrich Gauss, and for this reason, normal curves are sometimes known as Gaussian curves. Normal curves all share the same basic shape–that of a bell–but otherwise they can differ widely in their appearance.

Normal Curves


Some bells are short and squat,others are tall and skinny, and others fall somewhere in between.

Normal Curves


Mathematically speaking, however, they all have the same underlying structure. In fact, whether a normal curve is skinny and tall or short and squat depends on the choice of units on the axes, and any two normal curves can be made to look the same by just fiddling with the scales of the axes.What follows is a summary of some of the essential facts about normal curves and their associated normal distributions.These facts are going to help us greatly later on in the chapter.

Normal Curves


Symmetry.

Every normal curve has a vertical axis of symmetry, splitting the bell-shaped region outlined by the curve into two identical halves. This is the only line of symmetry of a normal curve, so we can refer to it without ambiguity as the line of symmetry.

Essential Facts About Normal Curves


Median / mean.

We will call the point of intersection of the


horizontal axis and the line of symmetry of the curve the center of the distribution.


Median / mean.

The center represents both the median M and the mean (average) of the data. Thus, in a normal distribution, M = . The fact that in a normal distribution the median equals the mean implies that 50% of the data are less than or equal to the mean and 50% of the data are greater than or equal to the mean.



In a normal distribution, M = .

(If the distribution is approximately normal, then M ≈ .)

MEDIAN AND MEAN OF A NORMAL DISTRIBUTION


Standard Deviation.

The standard deviation–traditionally denoted by the Greek letter (sigma)–is an important measure of spread, and it is particularly useful when dealing with normal (or approximately normal) distributions, as we will see shortly. The easiest way to describe the standard deviation of a normal distribution is to look at the normal curve.



Standard Deviation.

If you were to bend a piece of wire into a bell-shaped normal curve, at the very top you would be bending the wire downward.



Standard Deviation.

But, at the bottom you would be bending the wire upward.



Standard Deviation.As you move your hands down the wire, the curvature gradually changes, and there is one point on each side of the curve where


the transition from being bent downward to being bent upward takes place. Such a point is called a point of inflection of the curve.


Standard Deviation.The standard deviation of a normal distribution is the horizontal distance


between the line of symmetry of the curve and one of the two points of inflection, P´ or P in the figure.


In a normal distribution, the standard deviation equals the distance between a point of inflection and the line of symmetry of the curve.

STANDARD DEVIATION OF A NORMAL DISTRIBUTION


Quartiles.We learned in Chapter 14 how to find the quartiles of a data set. When the data set has a normal distribution, the first and third quartiles can be approximated using the mean and the standard deviation . The magic number to memorize is 0.675. Multiplying the standard deviation by 0.675 tells us how far to go to the right or left of the mean to locate the quartiles.



In a normal distribution,

Q3 ≈ + (0.675)and

Q1 ≈ – (0.675).

QUARTILES OF A NORMAL DISTRIBUTION


Imagine you are told that a data set ofN = 1,494,531 numbers has a normal distribution with mean = 515 and standard deviation = 114. For now, let’s not worry about the source of this data–we’ll discuss this soon.Just knowing the mean and standard deviation of this normal distribution will allow us to draw a few useful conclusions about this data set.

Example 16.3 A Mystery Normal Distribution


■ In a normal distribution, the median equals the mean, so the median value is M = 515. This implies that of the 1,494,531 numbers, there are 747,266 that are smaller than or equal to 515 and 747,266 that are greater than or equal to 515.



■ The first quartile is given by Q1 ≈ 515 – 0.675 114 ≈ 438.This implies that 25% of the data set (373,633 numbers) are smaller than or equal to 438.

■ The third quartile is given byQ1 ≈ 515 + 0.675 114 ≈ 592.

This implies that 25% of the data set (373,633 numbers) are bigger than or equal to 592.


16 mathematics of normal distributions

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